Question

If X is a vector, and A and B are linear operators; then the correct mathematical relationship(s) is/are

• A. (A+B)X = AX + BX
• B. (2A)X = 2(AX)
• C. (AB)X = A(BX)
• D. $(A+B)X = A^TX+B^TMX$

Answer 1

The Correct Option is A, B, C

Step 1: Define linear operators. Linear operators satisfy two properties for any vectors \( X, Y \) and scalar \( \lambda \): \( A(X + Y) = AX + AY \) (additivity) \( (\lambda A)X = \lambda (AX) \) (scalar multiplication) 

Step 2: Analyze each option. \( (A + B)X = AX + BX \): This is correct because the addition of two linear operators applied to \( X \) distributes over the addition. \( (\lambda A)X = \lambda (AX) \): This is correct because scalar multiplication of a linear operator applies directly to the result of \( AX \). \( (AB)X = A(BX) \): This is correct because the composition of two linear operators \( A \) and \( B \) acting on \( X \) satisfies this property. \( (A + B)X = A^T X + B^T X \): This is incorrect because the transpose (\( A^T \)) is not involved unless explicitly stated, and it does not apply to the given scenario.